Problem: Simplify and expand the following expression: $ \dfrac{3}{p + 6}+ \dfrac{4}{p + 1}+ \dfrac{3}{p^2 + 7p + 6} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{3}{p^2 + 7p + 6} = \dfrac{3}{(p + 6)(p + 1)}$ Now we have: $ \dfrac{3}{p + 6}+ \dfrac{4}{p + 1}+ \dfrac{3}{(p + 6)(p + 1)} $ The least common multiple of the denominators is: $ (p + 6)(p + 1)$ In order to get the first term over $(p + 6)(p + 1)$ , multiply by $\dfrac{p + 1}{p + 1}$ $ \dfrac{3}{p + 6} \times \dfrac{p + 1}{p + 1} = \dfrac{3(p + 1)}{(p + 6)(p + 1)} $ In order to get the second term over $(p + 6)(p + 1)$ , multiply by $\dfrac{p + 6}{p + 6}$ $ \dfrac{4}{p + 1} \times \dfrac{p + 6}{p + 6} = \dfrac{4(p + 6)}{(p + 6)(p + 1)} $ Now we have: $ \dfrac{3(p + 1)}{(p + 6)(p + 1)} + \dfrac{4(p + 6)}{(p + 6)(p + 1)} + \dfrac{3}{(p + 6)(p + 1)} $ $ = \dfrac{ 3(p + 1) + 4(p + 6) + 3} {(p + 6)(p + 1)} $ Expand: $ = \dfrac{3p + 3 + 4p + 24 + 3}{p^2 + 7p + 6} $ $ = \dfrac{7p + 30}{p^2 + 7p + 6}$